PHILOSOPHICAL TRANSACTIONS. 
XII. First Memoir on the Theory of Analytical Operations. By the Rev. R. Murphy, 
M.A. Fellow of Cains College, Honorary Member of various Philosophical 
Societies. Communicated by J. W. Lubbock, Esq. F.R.S. 
Received December 8, — Read December 22, 1836. 
§ i. 
i. The elements of which every distinct analytical process is composed are three, 
namely, first the Subject, that is, the symbol on which a certain notified operation is 
to be performed ; secondly, the Operation itself, represented by its own symbol ; 
thirdly, the Result, which may be connected with the former two by the algebraic 
sign of equality. 
Thus let a be the subject representing, we may suppose, some quantity, b the sym- 
bol for multiplication by b, and c the result or product ; for greater distinctness let the 
subject be inclosed in square brackets, the analytical process in this case is [a] b = c. 
Again, let x n be the subject, a symbol of operation denoting that x must be 
changed into x + h, and {x + h) n will evidently be the result, or 
[x n ] 4 = (x + h) n . 
Again, let a x be the subject, A a symbol of operation, which indicates that we are 
to subtract the subject itself from that which it becomes when x is changed into x + h, 
which is usually called taking the finite difference, then the result is 
* + h x 
a — a , 
or 
M A = M 0* - !)• 
Lastly, let d z denote the operation of taking the finite difference, and after dividing 
it by h, then putting h — 0, which is the same as finding the differential coefficient 
of the subject, which we may suppose represented by u, then 
2 B 
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